Question 54
Let $ABC$ a triangle and $I$ the center of his circle inscribed. the perpendicular to the line $(AI)$ passing through the point $I$ Cut the line $(AB)$ at one point $D$ and the line $(AC)$ at one point $E$. It is assumed that there are two points $F$ and $G$ On the segment $[BC]$ such as $BA = BF$ and $CA = CG$. Let $T$ the point of intersection of circles circumscribed to triangles $ADF$ and $AEG$. Show that the center of the circle circumscribes to the triangle $AIT$ is on the line $(BC)$.